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Monday, November 2, 2015

The relativization obstacle

I was wondering what was exactly the meaning of trying to compare
complexity classes relatively to some oracle, and, more precisely, how
such an approach may help to compare these classes without oracles. In this post, I will focus on the celebrated result by Baker, Gill and Solovay,
stating that:

(BGS) There exist oracles $A$ and $B$ such that $P^A = NP^A$ and $P^B \neq NP^B$.

This result imposes a strong constraint on any attempt to prove or refute $P ~=?~ NP$. Indeed, such a proof has to be non-relativizable.
Roughly speaking, a proof $\pi$ of a statement $S$ about complexity
classes is said to be relativizable if for any oracle $O$ the same proof
$\pi$ proves the statement $S^O$ which is $S$ with all complexity
classes being relativized to the oracle $O$. For example, if there were a
relativizable proof that $P = NP$, then we would have $P^O = NP^O$ for
all oracles $O$; which contradicts the BGS theorem above.

The lesson to retain is the following:

For any statement $S$ about complexity classes, if there exists an oracle $O$ such that $S^O$ holds (resp. does not hold), then there are no relativizable proofs that $S$ does not hold (resp. holds).

But what does a relativizable proof looks like ? Basically, it is a proof that combines composition of functions, and use of universal machines.

Let's try to prove that $P$ is a strict subset of $NP$ (sic!). More precisely, let's try to prove this claim using diagonal/simulation arguments as in the proof of the time hierarchy theorem. We could reduce (polynomial-time reductions) some $NP$-complete problem to some problem $H$ consisting of pairs $(i,n)$ such that the deterministic machine $i$ accepts input $n$ within $poly(|n|)$, plus, perhaps, additional properties. Then, we would assume that $H$ belongs to $P$, and try to derive a contradiction. We would have a deterministic polynomial-time machine $F$ which accepts $(i,n)$ if $(i,n)$ belongs to $H$, or rejects it otherwise. Then we would build a machine $K$ that satisfies the specification of $H$ on input $i$ if $F(i,i) = 0$, or does something contradicting the specification of $H$ otherwise. If $K$ on input $K$ satisfies the specification of $H$, then $F(K,K) = 0$, i.e., the machine $K$ on input $K$ does not satisfy the specification of $H$. If $K$ on input $K$ does not satisfy the specification of $H$, then $F(K,K) = 1$, i.e., the machine $K$ on input $K$ does satisfy the specification of $H$. Whence the 1'000'000 dollar contradiction.

The issue is, if we look at the proof above, we see that we can replace each Turing machine $M$ involved by the same machine $M^O$ augmented with some oracle $O$, without affecting the flow of arguments. In other words, such a proof is relativizing. The BGS theorem above prevents any such proof from solving the $P$ vs $NP$ problem. Bye bye dear 1'000'000 dollars.